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Theorem tpostpos2 7942
Description: Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.)
Assertion
Ref Expression
tpostpos2 ((Rel 𝐹 ∧ Rel dom 𝐹) → tpos tpos 𝐹 = 𝐹)

Proof of Theorem tpostpos2
StepHypRef Expression
1 tpostpos 7941 . 2 tpos tpos 𝐹 = (𝐹 ∩ (((V × V) ∪ {∅}) × V))
2 relrelss 6105 . . . 4 ((Rel 𝐹 ∧ Rel dom 𝐹) ↔ 𝐹 ⊆ ((V × V) × V))
3 ssun1 4062 . . . . . 6 (V × V) ⊆ ((V × V) ∪ {∅})
4 xpss1 5544 . . . . . 6 ((V × V) ⊆ ((V × V) ∪ {∅}) → ((V × V) × V) ⊆ (((V × V) ∪ {∅}) × V))
53, 4ax-mp 5 . . . . 5 ((V × V) × V) ⊆ (((V × V) ∪ {∅}) × V)
6 sstr 3885 . . . . 5 ((𝐹 ⊆ ((V × V) × V) ∧ ((V × V) × V) ⊆ (((V × V) ∪ {∅}) × V)) → 𝐹 ⊆ (((V × V) ∪ {∅}) × V))
75, 6mpan2 691 . . . 4 (𝐹 ⊆ ((V × V) × V) → 𝐹 ⊆ (((V × V) ∪ {∅}) × V))
82, 7sylbi 220 . . 3 ((Rel 𝐹 ∧ Rel dom 𝐹) → 𝐹 ⊆ (((V × V) ∪ {∅}) × V))
9 df-ss 3860 . . 3 (𝐹 ⊆ (((V × V) ∪ {∅}) × V) ↔ (𝐹 ∩ (((V × V) ∪ {∅}) × V)) = 𝐹)
108, 9sylib 221 . 2 ((Rel 𝐹 ∧ Rel dom 𝐹) → (𝐹 ∩ (((V × V) ∪ {∅}) × V)) = 𝐹)
111, 10syl5eq 2785 1 ((Rel 𝐹 ∧ Rel dom 𝐹) → tpos tpos 𝐹 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  Vcvv 3398  cun 3841  cin 3842  wss 3843  c0 4211  {csn 4516   × cxp 5523  dom cdm 5525  Rel wrel 5530  tpos ctpos 7920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-fv 6347  df-tpos 7921
This theorem is referenced by:  2oppchomf  17098  mattpostpos  21205
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